Question: The equation of hyperbola $H$ is $\dfrac{y^2}{49}-\dfrac {(x-5)^{2}}{64} = 1$. What are the asymptotes?
Solution: We want to rewrite the equation in terms of $y$ , so start off by moving the $y$ terms to one side: $\dfrac{y^2}{49} = 1 + \dfrac {(x-5)^{2}}{64}$ Multiply both sides of the equation by $49$ $y^2 = { 49 + \dfrac{ (x-5)^{2} \cdot 49 }{64}}$ Take the square root of both sides. $\sqrt{y^2} = \pm \sqrt { 49 + \dfrac{ (x-5)^{2} \cdot 49 }{64}}$ $ y = \pm \sqrt { 49 + \dfrac{ (x-5)^{2} \cdot 49 }{64}}$ As $x$ approaches positive or negative infinity, the constant term in the square root matters less and less, so we can just ignore it. $y \approx \pm \sqrt {\dfrac{ (x-5)^{2} \cdot 49 }{64}}$ $y \approx \pm \left(\dfrac{7 \cdot (x - 5)}{8}\right)$ Rewrite as an equality in terms of $y$ to get the equation of the asymptotes: $y = \pm \dfrac{7}{8}(x - 5)+ 0$